Fig. 2: Spatial representations emerge from aliased sequential random walk observations without Euclidean assumptions.

a A random walk in a room with only four unique observations (colors) will produce a severely aliased sequence as reflected in the first-order Markov chain. b In contrast, transition graph learned by CSCG on random walks in a recovers the spatial layout. Nodes in this graph are the clones, and the observation they connect to are indicated by the color of the node. c Room with a uniform interior produces aliased sequences highly correlated in time. d Transition graph learned by CSCG on random walks in c, represented similar to b. The redundant yellow nodes (and some brown nodes in b) are due to slight imperfections in learning, but do not affect the representation or behavior. e An agent experiences two different, but overlapping rooms in disjoint sequential episodes. The overlap region also repeats in the first room, acting as a confounder. f As reflected in the transition graph, CSCG performs transitive inference to stitch together the disjoint experience into a coherent global map, and correctly positions the confounder. g, h Activation of clones over time as the agent takes the trajectories X (gray), Y (black), and X again in the maze in g. Each red square is a clone activation in one time step. During the first traversal of X, the clones corresponding to the overlap and the confounding patch are active because the agent started within the overlap and stayed within. Stepping outside the overlap immediately resolves ambiguity, which is reflected in the clone activity during the traversal Y which includes confounder region and areas outside overlap and confounder, and also during the second traversal of X. See also Supplementary Video 1.